In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.
For part (b),
I have tried finding the Laplace transform of via the convolution property of Laplace transform.
My working is,
##L[\cos^2 (2t)] = L[\cos 2t] * L[\cos 2t]##
##L[\cos^2 (2t)] = \frac{s}{s^2 + 4} * \frac{s}{s^2 + 4}##
##\int_0^t \frac{s^2}{(s^2 + 4)^2} dt = \frac{ts^2}{(s^2 +...
Ok. My problem is what angle to choose when adding vector. Statement does not tell me which one is the "first" force vector. So, when using the law of sine formula I get two results.
First, using cosine to get the magnitude:
$$\vec c = \sqrt{a^2 + b^2 +2ab\cos\theta},$$
$$\vec c = \sqrt{15^2 +...
Going through this now: pretty straightforward i just want to check that i have covered all aspects including any other approach...
Ok for 15. I have,
##\nabla f= (yz \cos (xyz), xz \cos (xyz), xy \cos (xyz) )##
so,
##D_v f(1,1,1) = \textbf v ⋅\nabla f(1,1,1)##=##\left(\dfrac...
In my working i have,
...
##\cos C = 2\cos^2 \dfrac{1}{2} C -1##
##c^2= a^2+b^2-2ab(2\cos^2 \dfrac{1}{2} C-1)##
##c^2= a^2+b^2+2ab(1-2\cos^2 \dfrac{1}{2} C)##
##c^2= (a+b)^2 (1-2\cos^2 \dfrac{1}{2} C)##
Now from here, ##k^2 =2## but text gives different solution. I am still checking...
Problem statement : Let me copy and paste the problem as it appears in the text to the right.
Attempt :
Let me copy and paste my attempt. I couldn't go far, as you will see.
I couldn't progress from here. The powers of the ##\sin## and the ##\cos## are both what we want (##8##), but the...
In my approach (using a right angled triangle) i let,
##\cos^{-1} x = C## ⇒##\cos C = \sqrt{1-y^2}##
and
##\cos^{-1} y= A## ⇒ ##\cos A= \sqrt{1-x^2}##
Also, ##A+C = \dfrac{π}{2}##
and ##\cos \dfrac{π}{2}= 0##
##xy - \sqrt{(y^2) ⋅(x^2)}=xy-xy=0##
It follows that,
##\cos^{-1} [xy -...
Cosine error of a measured point on the surface is not a simple value, but a vector. If the ball contacts the part surface at a point located a distance from the theoretical or nominal point then the angle between the probe and the normal vector gets larger, P1 P2 will increase. We have cosine...
Hi,
The orthogonality defect is ##\prod_i ||b_i|| / det(B)##. Now it is said: The relation between this quantity and almost orthogonal bases is easily explained. Let ##\theta_i## be the angle between ##b_i## and ##span(b_1,...,b_{i-1})##. Then ##||b_i^*|| = ||b_i|| cos(\theta_i)##. [...]
So...
My question is; is showing the highlighted step necessary? given the fact that ##\sin (nπ)=0##? My question is in general i.e when solving such questions do i have to bother with showing the highlighted part...
secondly,
Can i have ##f(x)## in place of ##x^2##? Generally, on problems to do...
Hi, I am really struggling with the following problem on the Fourier sine and cosine transforms of the Heaviside unit step function. The definitions I have been using are provided below. I tried each part of the problem, but I'm only left in terms of limits as x -> infinity of sin or cos...
I need to find the zeros of this function where d,L,v are constants.
After several calculations I faced this equation.
I tried everything I know, but I can't solve this. Maybe I'm missing something or I must made a mistake earlier in the problem.
Thus, I would like to know if it is possible to...
Hi,
I have some question about evaluating a cosine function from ##-\infty## to ##\infty##.
I saw for a cosine function evaluate from ##-\infty## to ##\infty## I can change the limits from 0 to ##\infty##. I have a idea why, but I can't convince myself, furthermore, is it always the case no...
Given : The equation ##\sin m\theta + \sin n\theta = 0##.
Attempt : Using the formula for ##\text{sin C + sin D}## (see Relevant Equation 3 above), the given equation simplifies to
\begin{equation*}
2 \sin \frac{(m+n)\theta}{2} \cos \frac{(m-n)\theta}{2} = 0
\end{equation*}
This implies the...
Hi MHB! I recently came across a problem and I was thinking most likely I was missing something very obvious because I couldn't make sense of what was being asked, and I so wish to know what exactly that I failed to relate.
Question:
Find the minimum of $6\sin x+8\cos x+5$. Hence, find the...
A student has test his airplane and he is far from the airplane for 5 meter.He start to test his airplane by letting his airplane to move 60 degree from the horizontal plane with constant velocity for 120 meter per minute.Find the rate of distance between the student and the plane when the plane...
My function needs to speed up towards the left. How do I do this?
Green is the graph.
Red is my function that needs to match the graph.
A = Amplitude = -0.13
H = Phase Shift = 0.1625
V = Vertical Shift = 0.05
P = period = 0.4
B = 2π / P
Y = A (Cos(B...
Hello,
This is a more general question than anything, but I am curious how to compute the DFT of a cosine wave.
Somebody tried to explain this to me as follows:
start by trying to find an x(k) who's IDFT equals cos(2*pi*n/N).
x(k) = (N/2) * (dirac-delat(k+1) _ dirac-delta(k-1)) only has values...
Hello all,
In high school physics, the magnitude sum of vector addition can be found by cosine rule:
$$\vec {R^2} = \vec {F^2_1} + \vec {F^2_2} + 2 \cdot \vec F_1 \cdot \vec F_2 \cdot cos ~ \alpha$$
and its angle are calculated by sine rule:
$$\frac {\vec R} {sin ~ \alpha} = \frac {\vec F_1}...
I've already posted this question on the mathematics website of stack exchange, but I've received more help here in the past so will share it here as well.
I am developing a tool for missile trajectory (currently without guidance). One issue is that the aerodynamic equations on the missile are...
##cos(\omega)## is
$$\frac{e^{j \omega } + e^{-j \omega }}{2}$$
##sin(\omega)## is
$$\frac{e^{j \omega } - e^{-j \omega }}{2 j}$$
I also know that ##cos(\omega - \pi / 2) = sin(\omega)##.
I've been trying to show this using exponentials, but I can't seem to manipulate one form into the...
Continuing on from the summary, the chapter has given a graphed example. We are shown a regular cosine wave with phase angle 0 and another with phase angle (-Pi/4) in order to illustrate that the second curve is shifted rightward to the regular cosine curve because of the negative value. Now, my...
Hello all,
This is knowledge needed to solve my take-home final exam but I just want to ask about the definition of Bessel's integrals. This is not a problem on the exam. Wikipedia says the integral is defined as:
$$J_n(x) = \frac {1} {2\pi} \int_{-\pi}^{\pi} e^{i(xsin(\theta) - n\theta)} \...
How can the inequality ##cosx \ge(1-x^2/2)## be proved? Would you please explain how to prove this inequality?
This is the only equation that I could think of. ##1\ge cosx \ge 0## but I cannot use it here.
Source: Thomas's Calculus, this is from an integration question there.
Thank you.
1)* What are sine and cosine functions called in relation to Pi?
2) What is the exponential function called in relation to cosine and sine functions?
3) What are the other smooth, continual nested (or iterative) root functions (that are similar to sine and cosine) called in relation to...
Fourier Transform problem with f(t)=cos(at) for |t|<1 and same f(t)=0 for |t|>1. I have an answer with me as F(w)=[sin(w-a)/(w-a)]+[sin(w+a)/(w+a)]. But I can't show it.
Homework Statement
A line makes some angle T with each of x and z axis, and angle U with y-axis so that sin^2(U)=3sin^2(T).
Homework Equations
2cos^2(T)+cos^2(U)=1 ...(i)
cos^2(x)+sin^2(x)=1
The Attempt at a Solution
Using the above two eqns. give us the correct value for cos^2(T) which is 3/5...
The complex exponential form of cosine
cos(k omega t) = 1/2 * e^(i k omega t) + 1/2 * e^(-i k omega t)
The trigonometric spectrum of cos(k omega t) is single amplitude of the cosine function at a single frequency of k on the real axis which is using the basis function of cosine, right?
The...
Homework Statement
Take ∂2E/∂t2 E(r,t)=E0cos((k(u^·r−ct)+φ) in which u^ is a unit vector.
Homework Equations
d/dx(cosx)=-sinx
The Attempt at a Solution
I had calc 3 four years ago and can't for the life of me remember how to differentiate the unit vector. I came up with...
Homework Statement
A man walks 120m [S 23 E] Then turns and walks 60m [S 10 N] draw a careful sketch of this situation and then solve for the displacement, you do not have to solve for the direction
2. Relevant equation:
Cosine law
The Attempt at a Solution
Solution attempt is attached as a...
Homework Statement
Vectors A and B have equal magnitudes of 4.93. If the sum of A and B is the vector 6.79j, determine the angle between A and B
Homework Equations
c^2 = a^2 + b^2 -2abcos(theta)
The Attempt at a Solution
I just rearranged the formula above so that I could solve for the...
I'll start off with a given problem.
Find ##cos\left(α-β\right)## given that
##cos\left(α\right)=\frac{-12}{13}## and α is in quadrant III.
##sin\left(β\right)=\frac{-5}{13}## and β is in quadrant III.Solution:
##cos\left(α-β\right)=1##
This had we wonder if this continued for other angles of...
I have included a screenshot of a part of my textbook that is giving me a slight bit of confusion.
It's talking about how to get the formula for adding sines and cosines.
The part that I am confused about is the very first formula introduced in the screenshot.
From what I understand, we are...
Mentor note: Thread moved from technical section, so missing the homework template.
Hi all, I have a homework problem which asks me to compute the complex number cos(π/4 + π/4 i).
I've been playing around with it for a while now and just can't seem to get the answer I get when using Wolfram...
Homework Statement
[/B]
We have a function f(x) = |cos(x)|.
It's written that it is piecewise continuous in its domain.
I see that it's not "smooth" function, but why it is not continuous function - from the definition is should be..Homework Equations
[/B]
We say that a function f is...
Homework Statement
Homework Equations
All I know is the a's have something to do with the integrals.
The Attempt at a Solution
I used FFT analysis in Matlab but I do not know what I am looking for. How do the a0s relate to the f(t) in the question and how would I even do run that equation in...
Homework Statement
Cos(330 degrees)
*No calculator
Homework Equations
(a^2)+(b^2)=(c^2)
Cos=(delta x / hypotenuse)
The Attempt at a Solution
Hi guys, so today at school, the teacher was doing a problem which stated to solve the cosine of 330 degrees. The teacher used the conjugate angle...
Homework Statement
I am only interested in 9 (a)
Determine the Fourier Cosine series of the function g(x) = x(L-x) for 0 < x < L
Homework Equations
The Answer for 9 a.
g(x) = (L^2)/6 - ∑(L^2/(nπ)^2)cos(2nπx/L)
This is the relevant equation given where ω=π/L
f(t) = a0+∑ancos(nωt)
a0=1/L...
Homework Statement
-4.882L1+M1+2N1=0
L1-2.882M1=0
2L1-0.882N1=0
L1^2+M1^2+N1^2=1
How to solve for L1 ,M1 and N1 ?
0<L1,M1,N1<1Homework EquationsThe Attempt at a Solution
Hello all. After completing a problem in which we derived the formulas for potential and spring force energy as functions of time, with simple harmonic motion I noticed the equations are EXACTLY the same, but with sine and cosine switched. The equations were:
A sin^2(pi * t)
A cos^2(pi * t)
I...
1) Firstly, in the context of a dot product with Einstein notation :
$$\text{d}(\vec{V}\cdot\vec{n} )=\text{d}(v_{i}\dfrac{\text{d}y^{i}}{\text{d}s})$$
with ##\vec{n}## representing the cosine directions vectors, ##v_{i}## the covariant components of ##\vec{V}## vector, ##y^{i}## the...
Homework Statement
Homework Equations
cos2x = (1+cos2x)/2
sin2x = (1-cos2x)/2
The Attempt at a Solution
I believe you would use the double angle formula repeatedly but that is very tedious; is there a more concise way to solve the problem?
<Moderator's note: Moved from a technical forum and thus no template.>
Hello Forum,
This post is a spin-off from this post:
https://www.physicsforums.com/threads/how-much-would-time-pass-between-watching-the-sun-set.925257/#post-5840279
If I have an X and Y point on a grid that represents...
Homework Statement
Solve the equation
$$cos(\pi e^z) = 0$$Homework Equations
I am not allowed to use the complex logarithm identities.
$$ \cos z = \frac{e^{iz}+e^{-iz}}{2} $$
$$e^{i\theta}=\cos\theta+i \sin\theta$$
The Attempt at a Solution
All I've gotten is $$\cos(\pi e^z)=0 \iff \pi...